QUESTION:

Give me a brief formal definition of the ARIMA modeling process.

ANSWER:

B-J methods of analyzing time series data are based on the time domain tools of SACF and the SPACF.

B-J Methodology, consisting of several consecutive steps, allows one to fit a model to the observed data

and provide several methods of model testing.

B-J modeling consists of three steps:

     

  1. Identification:
    Using the tools such as SACF and SPACF identify a model to explain appropriately transformed data.

     

  2. Estimation:
    Estimate the parameters of the tentatively identified model from the previous step.

     

  3. Diagnostics:
    Check the adequacy of the model estimated in the previous step.

If the module is not satisfactory, we start with either a reduced model or an enhanced model based upon

the evidented structure in the residuals.

 

 

Model Identification

In order to identify a model we do the following:

Note that preliminary differencing may be necessary to obtain SACF and SPACF similar to

theoretical ACF and PACF of the ARIMA class model.

 

Box-Jenkins methodology looks for an adequate model in the group of models known as Auto Regressive

(AR), Moving Average (MA), Auto Regression Moving Average (ARMA) and Auto Regression Integrated

Moving Average (ARIMA) processes.

p order auto regressive process AR(p) is defined as:

X t = C + c 1 Xt - 1 + ... +c p Xt - p + at

where at is White Noise process. AR(p) is a stationary process if all roots of the characteristic polynomial,

1 - µ 1 X - ... - µp X p = 0

are greater than one in absolute value. An AR(p) process has the following distinguishing features.

q-order Average MA(q) is defined as:

X t = m + a t + é 1 at - 1 + ... +é q at - q,

where a t is a white noise process.

MA(q) is a stationary process.

An MA(q) process has the following distinguishing features.

The ARMA(p,q) process is defined as: X(t)-í(1)*X(t-1)-...-í(p)*X(t-p) = a(t)+é(1)*a(t-1)+..

.+é(q)*a(t-q),

where a(t) is a white noise process.